Modified Duration, Money Duration, and Price Value of a Basis Point (PVBP)

Modified Duration

Modified duration captures the sensitivity of a bond’s price to fluctuations in its yield-to-maturity (YTM). This relationship provides insight into how bond prices vary with shifts in yield. Specifically, bond prices and yields exhibit an inverse relationship: as yields rise, bond prices fall, and vice versa.

Relation to Macaulay Duration

Modified duration is an extension of the Macaulay duration, which conveys the weighted average time until a bond’s cash flows are received. The formula encapsulates the link between these two measures:

\[\text{ModDur} = \frac{\text{Macaulay Duration}}{1 + r}\]

Where \(r\) represents the yield per period. To obtain the annual modified duration, divide the modified duration by the bond’s number of coupon payments in a year. The larger the modified duration, the more pronounced the bond’s price-yield curve becomes, leading to larger price swings for given changes in yield.

Approximating Modified Duration

In cases where the Macaulay duration is not available, the modified duration can be estimated by observing minute variations in bond prices as yields change. This approximation method is especially useful for bonds with embedded options or inherent default risks. The formula for this approximation is:

\[\text{AnnModDur} \approx \frac{\left( PV_{-} – PV_{+} \right)}{2 \times \Delta\text{Yield} \times PV_{0}}\]

Where \(PV_{-}\)and \(PV_{+}\)are bond prices corresponding to decreased and increased yields, respectively. Historically, this method has been highly accurate. To revert to the Macaulay duration, multiply the modified duration by \(1 + r\).

Example: Approximating Modified Duration

A 4.5% semiannual-pay fixed-coupon bond is issued at par on 1 June 2026 and matures on 1 June 2030. For a 50bps increase and decrease in yield-to-maturity, \(PV_{+}\)and \(PV_{-}\)are 98.207 and 101.831, respectively. The approximate modified duration can be determined as follows:

Formula:

\[\text{AnnModDur} \approx \frac{\left( PV_{-} – PV_{+} \right)}{2 \times \Delta\text{Yield} \times PV_{0}}\]

\(PV_{-} =\)101.831

\(PV_{+}\) = 98.207

\(\Delta\text{Yield}\text{=50/10000=0.005}\)

\[AnnModDur\ \approx \frac{101.831 – 98.207}{2 \times 0.005 \times 100} = 3.624\ \]

Predicting Price Changes Based on Modified Duration

Modified duration unveils the bond price-yield relationship, allowing predictions of the bond’s percentage price alteration in relation to shifts in its YTM. The formula to determine this is:

\[\%\Delta PV^{\text{Full}} \approx – \text{AnnModDur} \times \Delta\text{AnnYield}\]

As an illustration, a bond with a modified duration of 5 would likely experience a \(5\%\) price drop if its yield surges by 100 basis points. Hence, bonds with higher modified durations exhibit steeper price-yield curves, making them more susceptible to yield variations. It’s crucial to note that this formula offers a linear approximation for the inherently nonlinear price-yield relationship. The inclusion of the negative sign emphasizes the inverse correlation between bond prices and their yields-to-maturity.

Money Duration

While modified duration gauges the percentage price change of a bond given variations in its yield-to-maturity (YTM), money duration provides insights into the price change in terms of currency units. In the U.S., it is also referred to as “dollar duration.”

Money duration is calculated using the formula:

\[\text{MoneyDur} = \text{AnnModDur} \times PV^{\text{Full}}\]

\(PV^{\text{Full~}}\ \)can be either the bond price as a percent of par value or the currency value of the bond holding.

Using Money Duration, one can estimate the bond price change in currency units for a given change in YTM:

\[\%\Delta PV^{\text{Full}} \approx – \text{MoneyDur} \times \Delta\text{Yield}\]

Example: Calculating Money Duration

Consider a bond with an annualized modified duration of 5.5, a coupon of \(4\%\), and a price of 102. The money duration is closest to:

\[\text{MoneyDur} = \text{AnnModDur} \times PV^{\text{Full}}\]

\[\text{Money Duration} = 5.5 \times 102\]

This means that for a \(1\%\) (or 100 basis points) change in yield, the bond’s price will change by $561.

Price Value of a Basis Point (PVBP)

PVBP provides an estimate of the change in the full price of a bond for a minuscule 1bp change in its YTM. PVBP can be determined using the formula:

\[PVBP = \frac{\left( PV_{-} \right) – \left( PV_{+} \right)}{2}\]

This measure is often termed as “PV01” or, in the U.S., “DV01” (Dollar Value of 1bp). PVBP is especially handy for bonds where future cash flows are unpredictable, like callable bonds.

Basis Point Value (BPV) is a close relative to PVBP, and it is the product of Money Duration and 0.0001 (1bp).

Question

An investment analyst is reviewing a 4-year bond issued on 1 January 2024 and set to mature on 1 January 2028. This bond features a 4% coupon rate, paid semi-annually, and carries a yield-to-maturity of 6%. The bond’s annualized Macaulay duration and Modified duration, respectively, are closest to:

1. 3.46 and 3.26.
2. 3.69 and 3.48.
3. 3.72 and 3.62.

Solution

The correct answer is C.

The Macaulay duration is 7.4481. This can be annualized by dividing by the number of coupon payments in a year.

\[ \begin{array}{c|c|c|c|c|c}\textbf{Period} & \textbf{Time to receipt} & \textbf{Cashflow amount} & \textbf{PV} & \textbf{Weights} & \textbf{Time to Receipt*Weight} \\ \hline 1 & 1.0000 & 2 & 1.9417 & 0.0209 & 0.0209 \\ \hline 2 & 2.0000 & 2 & 1.8852 & 0.0203 & 0.0406 \\ \hline 3 & 3.0000 & 2 & 1.8303 & 0.0197 & 0.0591 \\ \hline 4 & 4.0000 & 2 & 1.7770 & 0.0191 & 0.0764 \\ \hline 5 & 5.0000 & 2 & 1.7252 & 0.0186 & 0.0928 \\ \hline 6 & 6.0000 & 2 & 1.6750 & 0.0180 & 0.1081 \\ \hline 7 & 7.0000 & 2 & 1.6262 & 0.0175 & 0.1224 \\ \hline 8 & 8.0000 & 102 & 80.5197 & 0.8660 & 6.9279 \\ \hline \textbf{Total} & & & \textbf{92.9803} & \textbf{1.0000} & \textbf{7.4481} \\ \end{array} \]

\[Annualized\ Macaulay\ duration\ = \frac{7.4481}{2} = 3.72405\ \]

\[ModDur\ = \frac{3.72405}{1.03} = \ 3.6156\]